Abstract
In analogy to the role of Lommel polynomials in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form with parmeter v to Parabolic Cylinder functions Dv(z) is developed. The group-theoretical background with the 3-parameter group of motions M(2) in the plane for Bessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.
Highlights
Let us motivate our intentions by an analogy of Bessel functions Jν ( z ) to Parabolic Cylinder functions Dν ( z )
Our “Associated Hermite polynomials” are related to the Parabolic Cylinder functions in the analogous way as Lommel polynomials are related to Bessel functions
In Section we derive formulae for the differentiation of the Associated Hermite polynomials from which we develop the differential equation for these polynomials
Summary
Let us motivate our intentions by an analogy of Bessel functions Jν ( z ) to Parabolic Cylinder functions Dν ( z ). Both sets of functions satisfy a certain second-order differential equation and a certain 3-term recurrence relation. The. 3-term recurrence relation for the Bessel functions in the form Jν +n ( z ) pro-
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